Distinguish Rational and Irrational Values

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Which supplied value, among fractions, repeating decimals and roots of 4 or 5, is irrational? Candidate answers: (A) 0.222… with the block 2 repeating (B) 2/4 (C) √4 (D) √5.
• Independent check inputs: Which supplied value, among fractions, repeating decimals and roots of 9 or 10, is irrational? Candidate answers: (A) 0.333… with the block 3 repeating (B) 3/4 (C) √10 (D) √9.

Learn the idea

Roots reverse powers, but their exact values and decimal approximations must be distinguished. Integer squares provide reliable bounds, and odd roots retain negative signs while principal square roots are nonnegative. Fractions and terminating or repeating decimals are rational. A square root of a positive integer that is not a perfect square is irrational, even though decimal approximations can describe it.

Here is the method on the supplied teaching case. 5 lies between consecutive integer squares, so √5 is not a rational number. A fraction, a repeating decimal and the square root of a perfect square are rational.

A calculator display never proves irrationality by itself. Use the integer perfect-square test here; repeating decimals remain rational even when their written digits continue without end.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Take it further

Changed-input task: Which supplied value, among fractions, repeating decimals and roots of 81 or 82, is irrational? Candidate answers: (A) 0.999… with the block 9 repeating (B) 9/4 (C) √81 (D) √82. Show the mathematical reason, not only the selected answer. Expected result: √82. 82 lies between consecutive integer squares, so √82 is not a rational number. A fraction, a repeating decimal and the square root of a perfect square are rational.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.